Lipschitz modulus in convex semi-infinite optimization via d.c. functions

María J. Cánovas; Abderrahim Hantoute; Marco A. López; Juan Parra

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 15, Issue: 4, page 763-781
  • ISSN: 1292-8119

Abstract

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We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554]) constitute the starting point of the present work.

How to cite

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Cánovas, María J., et al. "Lipschitz modulus in convex semi-infinite optimization via d.c. functions." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2008): 763-781. <http://eudml.org/doc/90936>.

@article{Cánovas2008,
abstract = { We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554]) constitute the starting point of the present work. },
author = {Cánovas, María J., Hantoute, Abderrahim, López, Marco A., Parra, Juan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Convex semi-infinite programming; modulus of metric regularity; d.c. functions; convex semi-infinite programming; modulus of metric regularity},
language = {eng},
month = {7},
number = {4},
pages = {763-781},
publisher = {EDP Sciences},
title = {Lipschitz modulus in convex semi-infinite optimization via d.c. functions},
url = {http://eudml.org/doc/90936},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Cánovas, María J.
AU - Hantoute, Abderrahim
AU - López, Marco A.
AU - Parra, Juan
TI - Lipschitz modulus in convex semi-infinite optimization via d.c. functions
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/7//
PB - EDP Sciences
VL - 15
IS - 4
SP - 763
EP - 781
AB - We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554]) constitute the starting point of the present work.
LA - eng
KW - Convex semi-infinite programming; modulus of metric regularity; d.c. functions; convex semi-infinite programming; modulus of metric regularity
UR - http://eudml.org/doc/90936
ER -

References

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